MMath&Phys Mathematics and Physics

In Mathematics it is often useful to study mathematical objects (such as groups, vector spaces, and topological spaces) not only in isolation, but also in relation to each other, by considering the appropriate kind of morphisms between them (such as group homomorphisms, linear maps and continuous functions, respectively). This apparently simple idea led to the discovery of one of the most important concepts of 21st century mathematics, that of a category, and the development of the corresponding theory, Category Theory.

Category Theory allows us to establish precise analogies between different parts of mathematics and to discover unexpected connections between them. This led to deep applications in Algebra, Algebraic Geometry, Algebraic Topology, and Logic. As such, Category Theory should be of interest to a wide range of students interested in Pure Mathematics.

In this unit, you will first learn about the basic notions and results of Category Theory, leading up to adjunctions and limits. You will then see some applications of Category Theory and explore how fundamental notions of Algebra and Logic can be phrased and studied category-theoretically. Throughout the unit, definitions and theorems will be illustrated with concrete examples.

Some knowledge of Logic is beneficial but not necessary. 

MATH21120 Groups and Geometry and one of MATH21111 Metric Spaces or MATH21112 Rings and Fields

OR

MATH20201 Algebraic Structures 1 and one of MATH20212 Algebraic Structures 2 or MATH20122 Metric Spaces.  

Some knowledge of Logic is beneficial but not necessary. 

The unit aims to introduce the fundamental concepts, techniques, and results of Category Theory, and to illustrate how they can be applied to other parts of Mathematics, especially Algebra and Logic. 

• Define the notion of a category and check the axioms for it in simple examples.
   
• Define the notions of a functor and of a natural transformation and check the functoriality and naturality axioms in simple examples.
   
• Explain the equivalence between different formulations of the notion of an adjunction and apply them to identify adjunctions in simple examples.
   
• Name the fundamental kinds of limits and colimits and answer simple questions about their properties and relationship.
   
• Formulate and apply the fundamental result on preservation of limits by adjoint functors.
   
• Verify simple properties of syntactic categories.
   
• Relate precisely models of a theory and structure-preserving functors from its syntactic category.
   
• Relate precisely homomorphisms of models and natural transformations.
   

Syllabus:
Basic notions (6 lectures, 3 weeks). Categories (1 lecture). Initial and terminal objects, isomorphisms, monomorphisms, epimorphisms (1 lecture). Functors (1 lecture). Full and faithful functors (1 lecture). Natural transformations (1 lecture). Equivalence of categories (1 lecture).

Adjunctions and limits (8 lectures, 4 weeks). Adjunctions (1 lecture). Characterisation of adjunctions (2 lectures). Products, pullbacks, equalizers (1 lecture). Limits (1 lecture). Preservation of limits by adjoint functors (1 lecture). Duality (1 lecture). Colimits (1 lecture).

Functorial semantics (8 lectures, 4 weeks). Theories and their models (2 lectures). Syntactic categories (2 lectures). Functorial semantics (3 lectures). Outlook: dualities (1 lecture).
 

Feedback will be given on the weekly problem sheet assignments. Tutorials will provide an opportunity for students' work to be discussed and for feedback on their understanding to be given. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Final exam - Generic feedback made available after exam period 80%

Coursework (In-class test) - On returned scripts, within two weeks of examination 20%

S. Awodey, Category Theory, Oxford University Press (2nd edition), 2010.

T. Leinster, Basic Category Theory, Cambridge University Press, 2014.  

P. T. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Oxford University Press, 2002.

E. Riehl, Category Theory in Context, Dover Publications, 2016.